1*> \brief <b> ZHESV computes the solution to system of linear equations A * X = B for HE matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
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7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
22*                         LWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          UPLO
26*       INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            IPIV( * )
30*       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> ZHESV computes the solution to a complex system of linear equations
40*>    A * X = B,
41*> where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
42*> matrices.
43*>
44*> The diagonal pivoting method is used to factor A as
45*>    A = U * D * U**H,  if UPLO = 'U', or
46*>    A = L * D * L**H,  if UPLO = 'L',
47*> where U (or L) is a product of permutation and unit upper (lower)
48*> triangular matrices, and D is Hermitian and block diagonal with
49*> 1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
50*> used to solve the system of equations A * X = B.
51*> \endverbatim
52*
53*  Arguments:
54*  ==========
55*
56*> \param[in] UPLO
57*> \verbatim
58*>          UPLO is CHARACTER*1
59*>          = 'U':  Upper triangle of A is stored;
60*>          = 'L':  Lower triangle of A is stored.
61*> \endverbatim
62*>
63*> \param[in] N
64*> \verbatim
65*>          N is INTEGER
66*>          The number of linear equations, i.e., the order of the
67*>          matrix A.  N >= 0.
68*> \endverbatim
69*>
70*> \param[in] NRHS
71*> \verbatim
72*>          NRHS is INTEGER
73*>          The number of right hand sides, i.e., the number of columns
74*>          of the matrix B.  NRHS >= 0.
75*> \endverbatim
76*>
77*> \param[in,out] A
78*> \verbatim
79*>          A is COMPLEX*16 array, dimension (LDA,N)
80*>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
81*>          N-by-N upper triangular part of A contains the upper
82*>          triangular part of the matrix A, and the strictly lower
83*>          triangular part of A is not referenced.  If UPLO = 'L', the
84*>          leading N-by-N lower triangular part of A contains the lower
85*>          triangular part of the matrix A, and the strictly upper
86*>          triangular part of A is not referenced.
87*>
88*>          On exit, if INFO = 0, the block diagonal matrix D and the
89*>          multipliers used to obtain the factor U or L from the
90*>          factorization A = U*D*U**H or A = L*D*L**H as computed by
91*>          ZHETRF.
92*> \endverbatim
93*>
94*> \param[in] LDA
95*> \verbatim
96*>          LDA is INTEGER
97*>          The leading dimension of the array A.  LDA >= max(1,N).
98*> \endverbatim
99*>
100*> \param[out] IPIV
101*> \verbatim
102*>          IPIV is INTEGER array, dimension (N)
103*>          Details of the interchanges and the block structure of D, as
104*>          determined by ZHETRF.  If IPIV(k) > 0, then rows and columns
105*>          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
106*>          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
107*>          then rows and columns k-1 and -IPIV(k) were interchanged and
108*>          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
109*>          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
110*>          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
111*>          diagonal block.
112*> \endverbatim
113*>
114*> \param[in,out] B
115*> \verbatim
116*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
117*>          On entry, the N-by-NRHS right hand side matrix B.
118*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
119*> \endverbatim
120*>
121*> \param[in] LDB
122*> \verbatim
123*>          LDB is INTEGER
124*>          The leading dimension of the array B.  LDB >= max(1,N).
125*> \endverbatim
126*>
127*> \param[out] WORK
128*> \verbatim
129*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
130*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
131*> \endverbatim
132*>
133*> \param[in] LWORK
134*> \verbatim
135*>          LWORK is INTEGER
136*>          The length of WORK.  LWORK >= 1, and for best performance
137*>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
138*>          ZHETRF.
139*>          for LWORK < N, TRS will be done with Level BLAS 2
140*>          for LWORK >= N, TRS will be done with Level BLAS 3
141*>
142*>          If LWORK = -1, then a workspace query is assumed; the routine
143*>          only calculates the optimal size of the WORK array, returns
144*>          this value as the first entry of the WORK array, and no error
145*>          message related to LWORK is issued by XERBLA.
146*> \endverbatim
147*>
148*> \param[out] INFO
149*> \verbatim
150*>          INFO is INTEGER
151*>          = 0: successful exit
152*>          < 0: if INFO = -i, the i-th argument had an illegal value
153*>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
154*>               has been completed, but the block diagonal matrix D is
155*>               exactly singular, so the solution could not be computed.
156*> \endverbatim
157*
158*  Authors:
159*  ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \date November 2011
167*
168*> \ingroup complex16HEsolve
169*
170*  =====================================================================
171      SUBROUTINE ZHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
172     $                  LWORK, INFO )
173*
174*  -- LAPACK driver routine (version 3.4.0) --
175*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
176*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177*     November 2011
178*
179*     .. Scalar Arguments ..
180      CHARACTER          UPLO
181      INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
182*     ..
183*     .. Array Arguments ..
184      INTEGER            IPIV( * )
185      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
186*     ..
187*
188*  =====================================================================
189*
190*     .. Local Scalars ..
191      LOGICAL            LQUERY
192      INTEGER            LWKOPT, NB
193*     ..
194*     .. External Functions ..
195      LOGICAL            LSAME
196      INTEGER            ILAENV
197      EXTERNAL           LSAME, ILAENV
198*     ..
199*     .. External Subroutines ..
200      EXTERNAL           XERBLA, ZHETRF, ZHETRS, ZHETRS2
201*     ..
202*     .. Intrinsic Functions ..
203      INTRINSIC          MAX
204*     ..
205*     .. Executable Statements ..
206*
207*     Test the input parameters.
208*
209      INFO = 0
210      LQUERY = ( LWORK.EQ.-1 )
211      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
212         INFO = -1
213      ELSE IF( N.LT.0 ) THEN
214         INFO = -2
215      ELSE IF( NRHS.LT.0 ) THEN
216         INFO = -3
217      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
218         INFO = -5
219      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
220         INFO = -8
221      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
222         INFO = -10
223      END IF
224*
225      IF( INFO.EQ.0 ) THEN
226         IF( N.EQ.0 ) THEN
227            LWKOPT = 1
228         ELSE
229            NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 )
230            LWKOPT = N*NB
231         END IF
232         WORK( 1 ) = LWKOPT
233      END IF
234*
235      IF( INFO.NE.0 ) THEN
236         CALL XERBLA( 'ZHESV ', -INFO )
237         RETURN
238      ELSE IF( LQUERY ) THEN
239         RETURN
240      END IF
241*
242*     Compute the factorization A = U*D*U**H or A = L*D*L**H.
243*
244      CALL ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
245      IF( INFO.EQ.0 ) THEN
246*
247*        Solve the system A*X = B, overwriting B with X.
248*
249         IF ( LWORK.LT.N ) THEN
250*
251*        Solve with TRS ( Use Level BLAS 2)
252*
253            CALL ZHETRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
254*
255         ELSE
256*
257*        Solve with TRS2 ( Use Level BLAS 3)
258*
259            CALL ZHETRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
260*
261         END IF
262*
263      END IF
264*
265      WORK( 1 ) = LWKOPT
266*
267      RETURN
268*
269*     End of ZHESV
270*
271      END
272